MLE for Poisson Distribution¶
Overview¶
We derive the MLE for the Poisson distribution \(X \sim \text{Poisson}(\lambda)\) with PMF \(P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}\).
Derivation¶
The log-likelihood is:
\[
\ell(\lambda) = \left(\sum_{i=1}^n x_i\right) \log \lambda - n\lambda - \sum_{i=1}^n \log(x_i!)
\]
Setting the derivative to zero:
\[
\frac{d\ell}{d\lambda} = \frac{\sum x_i}{\lambda} - n = 0 \implies \hat{\lambda}_{MLE} = \bar{X}
\]
Properties¶
- \(\hat{\lambda}_{MLE} = \bar{X}\) is unbiased
- Fisher information: \(I(\lambda) = 1/\lambda\)
- \(\hat{\lambda}\) is efficient (achieves CRLB)